Approximation of spectroradiometric data by fractional model

Approximation of spectroradiometric data by fractional model

Dorota MOZYRSKA1, Irena FRYC2
Bialystok University of Technology, Faculty of Computer Science (1)
Bialystok University of Technology, Faculty of Electrical Engineering (2)
Approximation of spectroradiometric data by fractional model
Abstract. In the paper results of approximation of spectroradiometric data by fractional model are presented. Investigations of fractional parameters
show that they depend on the interval of lambda. By fractional model approximation we illustrate the theoretical basement for a model of an
emission intensity. For the development we use the fractional series model that comes from the similar model for energy.
Streszczenie. W pracy została dokonana analiza aproksymacji modelem potęgowym, z potęgą niecałkowitą, danych pochodzących z pomiaru
spektroradiometrycznego. Model opisu z potęgą niecałkowitego rzędu wykazuje zbieżność z teoretycznym opisem zjawiska. Użyto modelu szeregu
potęg niecałkowitego rzędu, który pochodzi z podobnego modelu dla zależności od energii. (Aproksymacja modelem potęgowym ułamkowego
rzędu rozkładu egzytancjii widmowej mierzonego promieniowania świetlnego)
Keywords: approximation, fractional model, spectroradiometric data.
Słowa kluczowe: aproksymacja, model potęgowy, pomiary spektroradiometryczne.
Introduction
In the paper we analyze the possibility of describing the
spectroradiometric measured data. The question is what an
approximation method and what type of a mathematical
model we can use. We discuss some privileged models. As
a shape of line seems to be Gaussian we checked the
goodness of fitting by that type of models. Moreover we
take under consideration also the nature of optical
properties of the emission intensity for an ideal LED, [1].
After analysis of the theoretical description we propose the
exact fractional power model that coincides with the
theoretical assumptions.
We use the approximation, not interpolation [2].
Methods of interpolation or approximation is widely
discussed in the literature, e.g. see [3]. However one can
find the discussion of sensitivity analysis of regression.
All calculations are done in MATLAB 7.5.0 (R2007b). As
the main criterion we take the minimum of the sum of
squares of errors – SSE and the same time the
maximization of the determination coefficient R-square. Such
parameters we obtain directly working with Curve Fitting
Toolbox in MATLAB program. We also point the important
case like confidence intervals for parameters. They are
given by 95% realization of confidence intervals. We do not
make any revision of approximation general methods.
Additionally one can find the description of methods and
guide of used commands in Matlab in description of Spline
Toolbox 3 or in Curve Fitting User’s Guide. Moreover some
basic commands are explored in the text. For Polish
readers we refer also [4,5,6].
Theoretical emission spectrum of an LED [1]
The physical mechanism by which semiconductor LEDs
emit light is spontaneous recombination of electron-hole
pairs and simultaneous emission of photons. The properties
of spontaneous emission in LEDs are discussed in [1]. In
our paper we only shortly describe the situation following
this book. The emission intensity as a function of energy is
proportional to the product of the joint density of states (E)
and the distribution of carriers in the allowed bands.
Moreover, according to [1], we have that
(1)
1  2mr* 
 (E)  2 
2  h / 2 2 
where h is Planck’s constant,
(2)
mr* 
1
1
 *
*
me mh
is the reduced mass for m*e and m*h being the electron and
hole effective masses, E g is the bandgap energy.
The distribution of carriers in the allowed bands is given
by the Boltzman distribution, i.e.
(3)
 E 
f B ( E )  exp 
.
 kT 
Then the emission intensity is written accordingly to
multiplication in Eqs. (1) and (3).
(4)
 E 
I ( E )  E  E g exp 
.
 kT 
In that case we easily see that I ( E ) is the multiplication of
power function of E  E g and exponential function for E .
Then theoretical emission intensity of an LED as the
function of E is presented in Fig. 1.
Fig. 1. Theoretical emission spectrum of an LED. The full width at
half maximum (FWHM) of the emission line is 1.8kT [1] .
The full width at half of the maximum of the emission is:
3/ 2
E  Eg ,
(5)
E  FWHM  1.8kT .
hc
, where c is the light velocity

and  is wavelength then the relation given by Eqs. (3) can
Let us notice that as E 
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 NR 2/2011
255
1
. It can be

analytically proved that the relation remains to be the power
type function of  . Then we claim that the emission
intensity given by Eqs. (4) we can rewrite as
be interpreted as the function of the inverse
(6)
Coefficients (with 95% confidence bounds):
a = 3.68e+131 (-2.311e+132, 3.047e+132)
b = -51.09 (-52.31, -49.86)
Goodness of fit:
SSE: 0.008155; R-square: 0.9925
 hc 
I ( )   hc  E g exp 
.
 kT 
The formula in Eqs. (6) is very sensitive in using it in
nonlinear approximation and it is dependent on E g .
However it can be investigated by using series expansion to
two one-sided infinite series of the following type
(7)



k
ak ( E g   ) 2 and



k
a k (  E g ) 2 .
But from another side we can also think about two-sided
approximation: one type of function up to the point of
maximum and the next function on the right of maximum. In
our investigations we can use a class of approximated
function as comes from the model of the emission intensity
given by formula in Eqs. (4).
Approximation for measurement LED data type
For a chosen measurement vector data type for LEDs
firstly we consider general Gaussian model. And applying
the curve fitting tool we get the following results for the
model its goodness.
General model Gauss2:
fittedmodel1(x) = a1*exp(-((x-b1)/c1)^2)
+ a2*exp(-((x-b2)/c2)^2)
Coefficients (with 95% confidence bounds):
a1 = 0.4571 (0.4537, 0.4605)
b1 =
379.8 (379.7, 379.8)
c1 =
10.07 (9.98, 10.15)
a2 = 0.001245 (-0.001518, 0.004008)
b2 =
506.9 (479.2, 534.6)
c2 =
15.27 (-23.87, 54.41)
Goodness of fit:
SSE: 0.01614; R-square: 0.9937
The plot of fitting curve is presented at the Fig. 2. (red line:
fitting curve, blue points: measurement points). From the
values of SSE and R-squared we see that the model is very
well fitted to the data. What it is only with doubt are some
points on the top excluded from the fitted curve.
Fig.2. Gaussian approximation function for LEDs data type.
As the next step we reconsider the half right part of the data
and the left part of data.
General model Power1:
f(x) = a*x^b
256
Fig. 3. Right part approximation function for LEDs data type for
lambda>380 nm.
General model:
f(x) = a*(385-x)^c
Coefficients (with 95% confidence bounds):
a = 1.761e+004 (8614, 2.66e+004)
c = -4.294 (-4.486, -4.102)
Goodness of fit:
SSE: 0.003924; R-square: 0.979
Fig.4. Left part approximation function for LEDs data type for
lambda<380 nm.
We also show by the example that considered model is
with agreement with measured data. For those purpose we
produce two vectors by excluding procedure in Matlab. One
for data with wavelengths longer than 380 nm and the
second with wavelengths not longer than 380 nm. And we
use the following models with calculated coefficients.
I) For wavelength greater than 380 nm we use the following
model:
General model:
f(x) =a+b*(x-380)^(0.5)+c*(x-380)^(-0.5)+d*(x-380)
+e*(x-380)^(3/2)+f*(x-380)^(-3/2)
Coefficients (with 95% confidence bounds):
a = -0.7023 (-0.7502, -0.6544)
b = 0.08275 (0.0741, 0.09141)
c=
2.078 (1.991, 2.166)
d = -0.004044 (-0.004656, -0.003432)
e = 7.012e-005 (5.561e-005, 8.463e-005)
f = -1.004 (-1.059, -0.9479)
Goodness of fit:
SSE: 0.01205
R-square: 0.9889
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 NR 2/2011
Conclusions
In all presented models we got very high value of
criterions. Considered data can be approximated by
fractional power model by one-sided function because of its
behavior. Series given by Equations (7) are used by
extracting few terms and then checking if the goodness of fit
is high enough. It is worthy to notice that also the
confidence intervals for coefficients are short. Our goal was
to confirm if from the model for energy we can introduce
also a model for the wavelength and if using measurement
data there exists an approximating function close to the
theoretical model.
Fig.5. Right approximation by the sum of power functions:
The work was done in Bialystok University of Technology
under grant S/WI/1/08.
Pracę wykonano w Politechnice Białostockiej w ramach
działalności naukowej S/WI/1/08.
f(x)=a+b*(x-380)^(0.5)+c*(x-380)^(-0.5)+d*(x-380)+e*(x380)^(3/2)+f*(x-380)^(-3/2) for lambda>380 nm.
REFERENCES
[1]
II) For wavelength less than 380 nm we use the following
model
General model:
f(x)=a+b*(380-x)^(0.5)+c*(380-x)^(-0.5)
+d*(380-x)+e*(380-x)^(3/2)+f*(380-x )^(-3/2)
Coefficients (with 95% confidence bounds):
a = -0.7667 (-1.529, -0.004206)
b = 0.2014 (0.06283, 0.34)
c = 0.3815 (-1.295, 2.058)
d = -0.01943 (-0.03078, -0.008083)
e = 0.0006445 (0.000298, 0.0009911)
f=
10.97 (7.721, 14.22)
Goodness of fit:
SSE: 0.001904
R-square: 0.9898
f(x) = a+b*(380-x)^(0.5)+c*(380-x)^(-0.5)+d*(380-x)
+e*(380-x)^(3/2)+f*(380-x )^(-3/2) for lambda<380 nm.
[2]
[3]
[4]
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Authors: Dorota Mozyrska, PhD, Bialystok University of
Technology, Faculty of Computer Science, Department of
Mathematics,
Wiejska
45a,
15-351
Białystok,
E-mail:
[email protected];
Irena Fryc, PhD, DsC, Bialystok University of Technology, Faculty
of Electrical Engineering, Department of Optical Radiation, Wiejska
45d, 15-351 Białystok, E-mail: [email protected];
Fig.6. Left approximation by the sum of power functions:
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